Interval uncertain optimization of structures using Chebyshev meta-models

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1 0 th World Congress on Strutural and Multdsplnary Optmzaton May 9-24, 203, Orlando, Florda, USA Interval unertan optmzaton of strutures usng Chebyshev meta-models Jngla Wu, Zhen Luo, Nong Zhang (Tmes New Roman, 0 pont, Bold, Centered) Shool of Eletral, Mehanal and Mehatron Systems,The Unversty of Tehnology, Sydney, NSW 2007, Australa zhen.luo@uts.edu.au Abstrat Ths paper proposes a new desgn optmzaton method for strutures subjet to unertanty. Interval model s used to aount for unertantes of unertan-but-bounded parameters. It only requres the determnaton of lower and upper bounds of an unertan parameter, wthout neessarly nowng ts prese probablty dstrbuton. The nterval unertan optmzaton problem wll be formulated as a nested double loop proedure, n whh both the desgn varables and strutural parameters are regarded as nterval numbers. In prate, the nested double loop optmsaton wll be omputatonally prohbtve, so the lnear optmzaton model s wdely used. However, the lnear optmzaton model ndues large error for strong nonlnear model. To mprove the auray and wthout nreasng omputaton ost muh, the nterval arthmet s appled to the nner loop to dretly evaluate the bounds of nterval funtons, replang the lnear model. The nterval arthmet s easly subjet to overestmaton due to ts ntrns wrappng effet, so the Taylor nluson funton wll be ntrodued to ompress the overestmaton n nterval omputatons. Sne t s hard to evaluate the hgh order oeffents n the Taylor nluson funton, a Chebyshev meta-model s proposed to appromate the Taylor nluson funton. A typal 25-bar spae truss struture optmzaton problem wth nterval unertantes are used to demonstrate the effetveness of the proposed method n the unertan desgn optmzaton of strutures. Keywords: struture optmzaton; nterval unertanty; Chebyshev meta-models; Taylor nluson funtons.. Introduton Desgn optmzaton of strutures has eperened onsderable development over the past two deades wth a wde range of engneerng applatons. However, the majorty of these wors are foused on the nvestgaton of the determnst optmzaton. In engneerng, there are many unertan fators nevtably related to materal propertes, geometry dmensons, loads and tolerane n the whole lfe yle of desgn, manufaturng, serve and agng of the struture [], due to the nherent unertan nature of the real-world systems. Hene, there s an nreasng demand to onsder the mpat of unertantes quanttatvely n the optmzaton of strutures n spte of unavodable varablty and unertanty, to enhane strutural safety and avod falure n etreme worng ondtons. To norporate unertantes n the desgn optmzaton, the determnst desgn problem should be sutably modfed and enhaned. The optmzaton wth unertanty manly ontans two paradgms, whh are the relable-based optmzaton (RBO) [2] and the robust desgn optmzaton (RDO) [3]. RDO ams at determnng a robust desgn to optmze the determnst performane about a mean value, whle mang t nsenstvty wth respet to unertan varatons by mnmzng the performane varane. RBO fouses on a rs-based soluton tang nto aount the feasblty of desgn target at epeted probablst levels, n whh the falure probabltes and epeted values are used to quanttatvely epress the effets of unertantes. In fat, RDO and RBO an be represented n the unform theory framewor. For nstane, Beyer and et al [4] also ndated that the RBO an be regarded as a spef ase of the RDO. In RDO and RBO methods, unertan parameters are mostly treated as random varables, wth prese probablty dstrbutons to be predefned based on the avalablty of omplete nformaton. However, t s generally a tme-onsumng and even an mpossble proess to aheve suffent unertan nformaton to determne probablty dstrbutons, due to the omplety of pratal problems [5]. Furthermore, Ben-Ham and Elshaoff [6] have shown that even small varatons devatng from real values may ause relatvely large errors of the probablty dstrbutons n the feasble regon. Hene, probablst methods may eperene dffulty for engneerng problems. At the same tme, there are a large number of desgn problems nvolved unertan-but-bounded parameters n engneerng. The nterval model has attrated muh attenton reently n the optmzaton of strutures [7]. In nterval models, the nterval number s used to measure the unertanty, beause the representaton of ntervals only requres bounds of unertan varables, whh an be determned easer than the probablty dstrbuton. The orrespondng bounds of an nterval funton are the mnmal and mamal responses of the unertan objetve and onstrants. The nterval method has been suessfully appled to the desgn of strutures nvolvng unertan-but-bounded

2 parameters [7]. Most onve models nvolve a nested double loop proedure. For nstane, a nested double-loop optmzaton method usng ellpsod models was proposed to strutural optmzaton [8], whh nluded the method of movng asymptotes n the outer loop and a sequental quadrat programmng n the nner loop. Although the nested double loop optmzaton s applable, the omputatonal ost s stll prohbtve, as eah ndvdual outer loop onssts of an nner loop mnmzaton. To redue the omputatonal ost of the nested optmzaton, the frst-order Taylor seres epanson has been appled to appromate the mamum or mnmum values of the bounds n the nner loop, nstead of usng the optmzaton algorthm. Kang and Luo [8] shown that a Taylor seres-based lnearzaton approah was more effent than a double loop optmzaton method, sne t an avod epensve teratons n the nner loop. However, ths method requres that the varablty of nterval varatons s relatvely small or moderate. Charaborty and et al [9] appled the matr perturbaton theory va a frst order Taylor seres epanson to obtan a onservatve dynam response of nterval funtons, also under the assumpton of a small level of unertanty. Chen and et al [0] used the frst-order Taylor seres epanson to analyse the robust response of nterval vbraton ontrol systems. In fat, the lnear model optmzaton s atually a type of degeneratve double loop optmzaton, n whh the nner optmzaton s replaed by the frst-order Taylor seres epanson (lnear model). However, the lnear model wth the low-order Taylor appromaton has a lower numeral auray and then the appromaton optmzaton may lead to a soluton n unfeasble regons. In ths researh, the nterval arthmet, whh defnes the fundamental arthmet operators, s ntrodued nto the nner optmzaton of the nested double loop proess to evaluate the mamum and mnmum values of an nterval funton, as the nterval arthmet an easly obtan the bounds of a desgn funton wth nterval parameters. However, the range of an nterval funton wll be enlarged n the numeral mplementaton, due to the nherent wrappng effet of the nterval arthmet. To ontrol the overestmaton, the Taylor nluson funton [] s utlzed to evaluate the bounds of nterval funton, and then the nterval arthmet s used to alulate the range of the polynomal funton. However, the oeffents, a set of hgh-order dervatves, n the polynomal funton s hard to be obtaned even for funtons wth eplt epressons. To ths end, the Chebyshev seres [2] are used to appromate oeffents of the Taylor nluson (polynomal) funton, so as to develop a Chebyshev meta-model. Ths meta-model an be onstruted by evaluatng funton values at some spefed nterpolaton ponts rather than the hgh-order dervatves, to mprove omputatonal effeny [2]. After obtanng the Chebyshev appromaton, the nterval arthmet an be used to alulate the bounds of the Taylor nluson funton n the nner loop. 2. Desgn optmzaton wth nterval unertantes A general determnst optmzaton model for the desgn of strutures s gven by mn f ( y, ) () s.t. g ( y, ) 0,, 2,..., n l u The above mathematal model s used to mnmze the objetve f subjet to onstrants g. R s the vetor q nludng determnst desgn varables, and y R s the vetor of onsstng of determnst parameters. To desrbe unertantes n the desgn, nterval numbers [] are ntrodued to epress the varatons ndued by the unertanty. Any nterval [] an be epressed as [ ] [, ] [ ]= +[-rad([ ]),rad([ ])] (2) where and denotes the lower and upper bounds of [], respetvely, ( + ) 2 denotes the mdpont of [], and [ ] denotes the symmetr nterval of [], rad([ ])= ( - ) 2denotes the radus refletng the unertan degree of []. Consder the unertantes, the determnst optmzaton model () an be re-defned as follows: mn f ([ ],[ y]) [] s.t. g ([ ],[ y]) 0,,2,..., n l u [ ] Here, the ranges for the nterval parameters [y] wll n general be pre-determned. Sne the radus of an nterval desgn varable [] s also pre-gven as ξ, any nterval desgn varable an be epressed as [ ]= +[- ξ, ξ ] (4) The responses of the objetve and onstrants would also be nterval numbers, denoted by [f] and [g], respetvely, beause the desgn varables and parameters are nterval vetors, (3) 2

3 f = f, f, g = g, g (5) The above mnmzaton problem s to mnmze both the average value and the wdth of the unertan objetve funton, to ensure the robustness of the desgn. The mnmzaton of the wdth wll lead to the derease of the varane of the objetve funton, to mae the unertan objetve funton nsenstve to the varaton due to the unertanty. It s noted that the mdpont value and radus are funtonally smlar to the probablst ounterparts n the onventonal robust desgn optmzaton, whh s a standard tehnque to mnmze both the mean value and the standard devaton of the objetve funton. To optmze the objetve, both the mdpont and radus of the objetve should be mnmzed, whh an atually be regarded as a type of robust desgns. Thus, the new objetve f an be spefed as obj f = f + rad([ f]) () obj where α and β denotes the weghtng oeffents, and we set both α and β as n ths study. Then the objetve an be re-defned as follows: fobj = f +rad([ f])= f (2) Then the objetve would be the upper bound of nterval [f], whh s the mamum value of f under the unertanty. For the nterval onstrants, there are three ases n the desgn spae: 0 g, g 0 g and. The frst ase volates the onstrant, and the seond ase ontans the possblty of volatng the onstrant. Only the last ase an guarantee the desgn ponts n the feasble regon, whh denotes a 00% relablty nde. So the upper bounds should be used to meet the onstrants g ([ ],[ y ]) 0,, 2,..., n (3) The upper bounds of the objetve and onstrants an be alulated through mamzng the value n the range of unertanty. Consder Eqs. (3), (7) and (8), the optmzaton model an be fnally epressed as mn ma f ( y, ) [ ], y[ y] s.t. ma g ( y, ) 0,, 2,..., n (4) [ ], y[ y] [ ]= +[- ξ, ξ], [ y]=[ y, y] l u + ξ -ξ 3. Lnear optmzaton model The optmzaton model n Eq. (9) nvolves a nested double loop optmzaton proess. The outer loop searhes the optmal mdpont of nterval desgn varables to mnmze the objetve, whle the nner loop fnds the mamum values (or mnmum values) of the objetve and onstrants wthn the ranges of nterval varables and parameters. The two optmzaton loops should use dfferent optmzaton strategy to balane numeral auray and omputatonal effeny, as the dfferent haratersts are possessed by the optmzaton models at two dfferent layers. To see the global optmal soluton and avod multple loal mnma, some heurst tehnques wth strong global searhng ablty have to be used. Ths study employs the Mult-Island Genet Algorthm (MIGA) [3] to solve the outer loop optmzaton problem. The MIGA s smlar to the general GA, whh onssts of two proesses: the frst proess s the seleton of ndvduals for the produton of the net generaton, and the seond proess s the manpulaton of the seleted ndvduals to produe the net generaton by rossover and mutaton tehnques. However, n MIGA, the populaton s dvded nto several sub-populatons and the mgraton operaton s added. Eah sub-populaton evolves ndependently for optmzng the same objetve funton. The mgraton ours every M generaton, and opes of the ndvduals whh are the best N% of the sland populatons are allowed to mgrate. M s alled the nterval of the mgraton and N% s alled the rate of the mgraton [3]. In the mgraton, the top N% strngs n the sub-populaton A may be oped to another sub-populaton B, and the least N% strngs of the sub-populaton B wll be elmnated. Smlarly, the sub-populaton A wll reeve the top strngs from other sub-populaton and elmnates ts least strngs. Ths operaton repeats untl eah sub-populaton aheves top strngs from another sub-populaton. Although MIGA has hgher possblty to searh the global optmum, ts onvergene rato s slower than tradtonal gradent-based algorthms, espeally wthn the neghbourhood of the optmal soluton. To mprove the effeny, the Sequental Quadrat Programmng (SQP) s nluded to searh the optmal pont after MIGA, whh means the optmal pont of MIGA s used as the ntal pont of SQP. In ths ase, the number of generatons n MIGA an be redued, beause only a lmted number of ponts near the global optmal soluton are requred, whh wll greatly derease the alulaton tme of MIGA. 3

4 The desgn spae of the nner loop optmzaton s relatvely narrow, so the nner loop an be appromated by lnear model. That s, the objetve an be epressed wth respet to the desgn varables and parameters at the mdponts va the frst-order Taylor seres f f f y, = f, y+ - + yy - +O2 (0) y (, y) (, y) If the hgher order terms are gnored, the mamum value an be determned as f ma y, f, y + rad [ ] + rad [ y] f f y (, y) (, y) where rad([ ]) and rad([ y]) denote the radus of nterval varables [] and parameters [y], respetvely. Smlarly, the mamum value of onstrant funtons g an also be alulated va the lnear model as g ma g g y, g, y + rad [ ] + rad [ y] y, y, y The trunated errors et n ths lnear model, due to the neglet of the hgher-order terms n Eq. (0). For hgh nonlnear problems, the trunaton error annot be gnored. The lnear model an mprove omputatonal effeny of the double loop optmzaton. However, t may result n a poor numeral auray due to the trunated error. To redue the omputatonal tme wthout sarfng numeral auray, a new optmzaton strategy wll be proposed based on the nterval arthmet. The flowharts for the lnearzed optmzaton usng the frst-order Taylor seres s shown n Fg. The lower-order Taylor seres epanson s used to replae the nner loop, so as to avod the omputatonally epensve double loop proess. Ths study wll employ an alternatve method to aheve a balaned performane of the effeny and numeral auray n the numeral mplementaton. () (2) Fgure : Flowhart of lnear optmzaton model 4. Interval optmzaton usng Chebyshev meta-models In ths seton, the proposed methodology nludes three parts: () the nterval arthmet s ntrodued to alulate the bounds of nterval funtons, to elmnate the nner optmzaton, (2) the Taylor nluson funton wth hgher-order seres s used to redue the overestmaton trggered by the wrappng effet, whh s ntrns n nterval arthmet. However, hgher-order dervatves are nvolved as oeffents n the alulaton of the Taylor nluson funton, whh agan weghts the omputatonal ost, and (3) to ths end, a Chebyshev meta-model s proposed to appromate the Taylor nluson funton, to mprove the effeny and auray. 4. Taylor nluson funton n the nterval arthmet The notaton of nterval numbers has been ntrodued n Seton 2. The nterval arthmet defnes some bas arthmet operatons between two dfferent nterval numbers. Consder two nterval varables [] and [y], and the bas arthmet operatons [] between them an be defned as follows: 4

5 [ ] [ y] y, y, [ ] [ y] y, y, (3) [ ] [ y] mn y, y, y, y,ma y, y, y, y, [ ] [ y] mn y, y, y, y,ma y, y, y, y, f 0 [ y] From Eq. (4), t an be found that the nterval arthmet only depends on the bound of nterval varables, whh an obvously mprove the omputatonal effeny of the optmzaton problem. However, nterval arthmet wll lead to a large overestmaton n the optmzaton, beause of the dependene between nterval varables. The nterval funton [f] s an nluson funton of the funton f f [ ] IR n, f ([ ]) [ f ]([ ]) (4) For a large lass of funtons f, one of the most mportant objetves for the nterval analyss s to provde nluson funtons [f] for f. Ths s requred to be evaluated reasonably, suh that the result s not too large. The natural nluson funton, whh s the produt by dretly applyng nterval arthmet to evaluate nterval funtons, wll often result n relatvely large overestmaton due to the wrappng effet of ntervals. To mae the result sharper, the hgh-order Taylor seres epansons of funtons are usually used. If the funton f s (n+) tmes partally dfferentable on an openng set ontanng the nterval [], the nth-order Taylor nluson funton an be epressed as ( n) n ( n) n f... T f n f f f (5) n! n! The front n+ terms at the rght sde of Eq. (5) are the trunated Taylor seres epanson of f(). The Eq. (5) alulates the rgorous enlosure for the funton f(). In general, the last term at the rght hand sde of Eq. (5) an be negleted to obtan the appromate enlosure of f(). Etendng Eq. (5) to -dmenson problem and negletng the remander terms, we an obtan the mult-dmensonal Taylor nluson funton [ f ]([ ]) [ ]...[ ] (6) where n... denote the oeffents whh are related wth the partal dervatves of f wth respet to, and the total number of oeffents s NT =( n+ )! n!!. In most ases, the Taylor nluson funton wll produe a narrower nterval than the nterval alulated dretly by nterval arthmet. However, a major problem of the hgher-order Taylor nluson funton s that a set of hgh-order partal dervatves, atng as the oeffents of the evaluaton funton, are requred to be alulated. Sne the hgh-order dervatves are hard to alulate, another numeral method wll be appled to evaluate these oeffents, whh wll lead to a meta-model for the appromaton of the hgh-order Taylor nluson funton. 4.2 Chebyshev meta-model The Chebyshev seres an also be used to epand the ontnuous funton, whh may produe hgher auray than Taylor seres epanson. Wu and et al [2] has shown that the Chebyshev polynomals have hgher appromaton auray than the Taylor polynomals under the same orders. To smplfy the problem but wthout losng any generalty, we onsder a varable [-, ]. The ontnuous funton f() an be appromated by n n f( )... f... C... ( ) (7) p where p denotes the total number of zero(s) to be ourred n the subsrpts,...,. C ( )... s the -dmensonal Chebyshev polynomals [2], f s a th-order tensor wth (n+) elements. Eah oeffent of the Chebyshev... polynomals an be alulated usng the followng ntegral formula [2]: ( )... ( ) m m 2 f C 2 f (,..., )... (,..., ) d d f j j C j j (8)... m j j where m denotes the order of numeral ntegral formula (m=n+ n ths study), j are the nterpolaton ponts of numeral ntegral formula. The nterpolaton ponts n eah dmenson are the zeros of (n+)th order Chebyshev polynomal, to be determned by 2j j os j, where j, j,2,..., n+ (9) n 2 Thus, the number of nterpolaton ponts for a -dmensonal problem would be N s =(n+). 5

6 From Eqs. (7) to (9), t an be found that the proess of onstrutng the Chebyshev appromant s smlar to the response surfae methodology (RSM), whh obtans the data at samplng ponts (or nterpolaton ponts n ths study) and then produes the oeffents based on these data. Equaton (7) an be transformed to a polynomal based on the power funton n n p n n f ( )... f... C... ( ) F (20) 0 0 where F denotes the oeffents after the transformaton. In Eq. (20), replang the varable wth the nterval... varable [], we an obtan the Taylor nluson funton. However, Eq. (20) ontans (n+) terms, whle the number of tems n the Taylor nluson funton (Eq. (6)) s NT =( n+ )! n!! whh s usually smaller than (n+). Thus, f the Chebyshev polynomals are used to appromate the Taylor nluson funton, some hgher order tems wll not be neessary. At the same tme, the number of nterpolaton ponts for onstrutng Chebyshev polynomal equals to the number of tems n Eq. (6), whh s stll omputatonally epensve, espeally for the hgh dmensonal problems. To further save the omputatonal ost, only a part of the nterpolaton ponts wll be used to buld the Chebyshev polynomals, whh s termed Chebyshev meta-model. Removng the tems wth orders hgher than n n Eq. (20), the meta-model an be epressed by p n n f ( ) f C ( ) = F... (2) Sne only a part of nterpolaton ponts are used to onstrut the Chebyshev meta-model, the Eq. (8) annot be used to alulate the oeffents. To redue the error between the meta-model and evaluaton funton, the least squares method (LSM) an be employed to produe the oeffents. The number of oeffents n Eq. (2) s N T, so the number of samplng ponts from the nterpolaton should not less than N T. At the same tme, the number of nterpolaton ponts s N s =(n+), whh s larger than N T when >. Therefore, the number of samplng ponts an be hosen as any number n the nterval [N T, N s ]. The larger number of samplng ponts, the smaller error of the appromaton, but lower effeny. Some studes [4] show there wll be a good balane between the auray and effeny, when the number of the samplng ponts s twe of the number of the oeffents. Thus, when N s >2 N T, the 2 N T nterpolaton ponts are hosen as the samplng ponts randomly. Otherwse, all the nterpolaton ponts are hosen as the samplng ponts. After the set of samplng data s obtaned, the LSM s used to alulate the oeffents and establsh the meta-model. After the Chebyshev meta-model s obtaned, t an then be ombned wth the outer loop optmzaton (MIGA+SQP) to mplement the unertan optmzaton. The major advantage of the nterval arthmet s that the mamum and mnmum values of a funton are ontaned n the nterval results, whh provde rgorous onstrants for the outer loop to guarantee the outer loop optmal soluton s n the feasble regon. The optmal desgn of the nterval arthmet may be more onservatve than that of the double loop optmzaton, but t s more relable than the double loop optmzaton. The flowhart n Fg. 2 llustrates the numeral proess of the proposed nterval optmzaton strategy, whh use the Chebyshev meta-model to replae the lnear model. Startng pont 0,y 0, =0 Construt the Chebyshev meta-model; Calulate f ma (,y ), g ma (,y ) usng nterval arthmet; Update desgn vaables:, y (MIGA) =+ No Converge? End Yes Fgure 2: The flowhart of nterval optmzaton strategy 5. Numeral eamples 6

7 Fgure 3 shows the 8-bar the 25-bar truss struture for transmsson towers [5] has been wdely studed, mostly under the assumpton that all the varables are determnst. In ths study, we onsder the ross-setonal areas of truss members as nterval varables, and the nterval wdth s 0.n 2. The densty of materal s 0.lb/n 3, and the elastty modulus s 0 7 lb/n 2. The objetve s to mnmze the total weght of the spae truss. Ths spae truss s subjeted to two loadng ondtons, whh are shown n Table. The struture s requred to be doubly symmetr about - and y-as, and so the truss members an be grouped as Table 2, whh also shows the stress lmtatons of eah group. At the same tme, the mamum dsplaements of nodes n eah dreton are lmted to ±0.35n. The ross-setonal area of all members s hanged n the range of 0.0~0n 2. Fgure 3: 25-bar spae truss struture The unertan optmzaton model an be defned as 25 mn ma w AL s.t. g = ma ma 40000; g2= mn mn 35092;,...,25,0,...,3 g 3= mn mn 590; g4= mn mn 7305; 2,...,5 6,...,9 g 5= mn mn 6759; g6= mn mn 6959; 4,...,7 8,...,2 g 7= mn mn 082; g8= ma ma d 0.35; 22,...,25,...,30 T ξ ξ,ξ 8 T T 4 4 ξ ξ where denotes the stress of th member, and d s the dsplaement of eah node n eah dreton. Table : Loadng ondtons Node Condton Condton 2 P (lb) P y (lb) P z (lb) P (lb) P y (lb) P z (lb) The lnear model optmzaton and nterval optmzaton methods are appled to solve ths problem, respetvely. The results n n Table 3 show that the two methods an result n smlar objetve funton values (rangng from 63 to 64). (22) 7

8 Table 2: Member stress lmtatons varables A A 2 ~A 5 A 6 ~A 9 A 0 A 2 A 4 A 8 Compressve stress lmtatons (lb/n2) Tensle stress lmtatons (lb/n 2 ) ~A ~A 3 ~A 7 ~A 2 A 22 ~A Table 3: The optmzaton results (n 2 ) 2 (n 2 ) 3 (n 2 ) 4 (n 2 ) 5 (n 2 ) 6 (n 2 ) Lnear optmzaton Interval optmzaton (n 2 ) 8 (n 2 ) g 6 (lb/n 2 ) g 8 (lb) W(n) Tme Lnear optmzaton (-6974) (0.354) Interval optmzaton (-6695) (0.3499) However, aordng to the valdated values of the onstrants shown n the braets, t an be found that the lnear optmzaton volates the two onstrants, whh are g 6 =-6974 lb/n 2 and g 8 =0.354n. The values mared by red olour wth underlnes are used to denote the volaton of the onstrants. The nterval optmzaton s loated n a safer poston for both of the onstrants (the rest 6 onstrants, whh are not lsted n ths table, satsfy the gven ondtons). Thus, the nterval method would eep the optmzaton result le n a feasble regon but the lnear method often generates more rs results. For the alulaton tme, the nterval optmzaton method taes 3 30, whh s aeptable relatve to the lnear optmzaton method Conlusons Ths researh has proposed a new unertan optmzaton method for the desgn of strutures nvolvng unertan-but-bounded parameters. The nterval model s used to desrbe unertantes of the bounded parameters, whh only requres the lower and upper bounds of an nterval number. The proposed nterval unertan optmzaton model has the haratersts of both the robust desgn and relablty based optmzaton. The nterval optmzaton ommonly leads to the nested double loop proess. In the lnear optmzaton proedure, the outer loop s usually used to update the desgn varables to see the optmal soluton whle the lnear model (nner loop) s to alulate the bounds of the nterval objetve and onstrants. The lnear optmzaton model has hgher effeny than tradtonal nested double loop optmzaton proess, but lower auray. To mprove the auray of the lnear optmzaton model, the nterval arthmet has been ntrodued nto the nner loop to dretly evaluate the bounds of nterval desgn funtons, so as to replae the lnear model. Furthermore, to redue the overestmaton n the nterval arthmet, the hgh-order Taylor nluson funton s utlzed to alulate the bounds of the nterval desgn funtons. However, the alulaton of the hgh-order dervatves n the nluson funton s not easy. Hene, the Chebyshev meta-model s norporated n the nluson funton to appromate the hgh-order dervatves, so that a Chebyshev model s developed, whh an provde hgher appromaton auray than the trunated Taylor seres. Typal numeral eamples are used to demonstrate the effetveness of the proposed nterval optmzaton methodology. Compared to the lnearzed optmzaton, the method n ths paper an mprove numeral auray and relablty wthout nreasng omputaton ost largely. 7. Referenes [] G.I. Shuëller, H.A.Jensen, Computatonal methods n optmzaton onsderng unertantes - an overvew, Computer Methods n Appled Mehans and Engneerng 98 (2008) 2-3. [2] M. Papadraas, N.D. Lagaros, Relablty-based strutural optmzaton usng neural networs and Monte Carlo smulaton, Computer Methods n Appled Mehans and Engneerng 98 (2008) [3] M.A. Valdebento, G.I. Shuëller, A survey on approahes for relablty-based optmzaton, Strutural and Multdsplnary Optmzaton 42 (200) [4] H.G. Beyer, B. Sendhoff, Robust optmzaton a omprehensve survey, Computer Methods n Appled Mehans and Engneerng 96 (2007) [5] B. Möller, M. Beer, Engneerng omputaton under unertanty - apabltes of non-tradtonal models. Computers and Strutures 86 (2008) [6] Y. Ben-Ham, I. Elshaoff, Conve models of unertantes n appled mehans. Amsterdam: Elsever Sene Publsher (990). 8

9 [7] C. Jang, X. Han, G.R. Lu, Optmzaton of strutures wth unertan onstrants based on onve model and satsfaton degree of nterval, Computer Methods n Appled Mehans and Engneerng 96 (2007) [8] Z. Kang, Y. Luo, Relablty-based strutural optmzaton wth probablty and onve set hybrd models, Strutural and Multdsplnary Optmzaton 42 (2009) [9] S. Charaborty, B.K. Roy, Relablty based optmum desgn of tuned mass damper n sesm vbraton ontrol of strutures wth bounded unertan parameters, Probablst Engneerng Mehans 26 (20) [0] S. Chen, M. Song, Y. Chen, Robustness analyss of responses of vbraton ontrol strutures wth unertan parameters usng nterval algorthm, Strutural Safety 29 (2007) 94-. [] L. Jauln, Appled nterval analyss: wth eamples n parameter and state estmaton, robust ontrol and robots, Sprnger, New Yor, 200. [2] J. Wu, Y. Zhang, et al., Interval method wth Chebyshev seres for dynam response of nonlnear systems, Appled Mathematal Modellng 37 (203) [3] D. Whtley, S. Rana, R.B. Heendorn, The Island Model Genet Algorthm: On separablty, populaton sze and onvergene, Journal of Computng and Informaton Tehnology 7 (998) [4] S.S. Isuapall, Unertanty analyss of transport-transformaton models, n, The State Unversty of New Jersey, New Brunsw, New Jersey, 999. [5] W. Gao, Natural frequeny and mode shape analyss of strutures wth unertanty, Mehanal Systems and Sgnal Proessng 2 (2007)